Information Dynamics in Evolving Networks Based on the Birth-Death Process: Random Drift and Natural Selection Perspective

TL;DR

This work develops a birth-death evolving network (BDEN) framework with two attachment schemes (uniform and preferential) to study information diffusion under turning populations. It establishes stationary network properties, notably $\mathbb{E}[N]=\lambda/\mu$ and, for BDEN-uc, $\mathbb{E}[k]=m$, and formulates two diffusion paradigms—random drift and natural selection—each leading to absorbing homogeneous states; fixation analyses reveal how parameters like $\lambda$, $\mu$, $m$, and transmission rate $\alpha$ govern information spread. Simulations validate theoretical predictions and show that random drift is more sensitive to attachment rules than natural selection, while starting from real networks yields similar cooperative thresholds. The findings provide a probabilistic, game-theoretic lens on information dynamics in populations with turnover, with implications for contagion, cooperation, and cultural evolution in evolving networks.

Abstract

Dynamic processes in complex networks are crucial for better understanding collective behavior in human societies, biological systems, and the internet. In this paper, we first focus on the continuous Markov-based modeling of evolving networks with the birth-death of individuals. A new individual arrives at the group by the Poisson process, while new links are established in the network through either uniform connection or preferential attachment. Moreover, an existing individual has a limited lifespan before leaving the network. We determine stationary topological properties of these networks, including their size and mean degree. To address the effect of the birth-death evolution, we further study the information dynamics in the proposed network model from the random drift and natural selection perspective, based on assumptions of total-stochastic and fitness-driven evolution, respectively. In simulations, we analyze the fixation probability of individual information and find that means of new connections affect the random drift process but do not affect the natural selection process.

Figures (9)

• Figure 1: An example of the evolving network model. This figure shows an example of the evolving networked population based on the birth-death process. The vertices in the initial network are in orange, where the corresponding vertex set is $V(0)=\{a, b, c, d, e, f\}$. New individuals, including $g, h, i, j$, arrive at the network and connect to the existing vertices by Poisson process at $t_1, t_3, t_4, t_6$ respectively. At $t_2$ and $t_5$, the vertices $e$ and $f$ leave the system and disconnect all their neighbors. For instance, the vertex set and edge set at $t_2$ are $V(t_2)=\{a, b, c, d, f, g\}$ and $E(t_2)=\{ab, ac, ag, bf, bd, cg, df\}$. (Color online)
• Figure 2: An example of the information dynamic model. This figure shows an example of the information diffusion process on the mentioned evolving network model. Vertices in blue and yellow denote the two states among the population. Here, when a new individual is arriving, it plays the same strategy as the blue nodes with the probability $2/3$, meaning that newcomers are more likely to be attracted to the population with the mainstream information in our model. After connecting to existing vertices, this new individual is affected by surroundings, which helps to adapt to the environment. (Color online)
• Figure 3: Results on network sizes. (a) The evolution of network sizes from $t=0$ to $t=10^4$. (b) The comparison of the theoretical results to the simulations on the expected network sizes. (c) The frequency of network sizes in the evolving process. These results hold for both BDEN-uc and BDEN-pa. Here, we set $\lambda$s$=[2,3,4,5]$, $m=5$, and the initial network size as $N(0)=30$ for all results. In (a), we set $\mu=0.01$. The $x$-axis and $y$-axis are set as the time $t$ and the network size $N(t)$. Dashed lines present the expectation $E[N]=\lambda/\mu$. In (b), the $x$-axis is set as the leave rate $\mu\in[0.005,0.02]$, and the simulations are carried out with the interval $0.001$. The $y$-axis is set as the average network size. Each data point in triangle is obtained by averaging the network sizes from $t=7\times10^3$ to $t=10^4$. Dashed lines are the theoretical solutions of expected network sizes. In (c), we set the $x$-axis as the network size $N$, and the $y$-axis as the corresponding probability in the evolving process. The samples are the network sizes from $t=10^3$ to $t=10^4$. (Color online)
• Figure 4: Results on mean degrees. (a) The evolution process of mean degrees with the fixed new connection number $m$. (b) The evolution process of mean degrees with the fixed input rate $\lambda$. (c) The curves of mean degrees against $\lambda$s. (a), (b), and (c) are for BDEN-uc, and (d), (e), (f) are for BDEN-pa with the same other settings. In this figure, we set the leave rate of each individual as $\mu=0.01$ and the initial network size $N(0)=30$. In (a) and (d), we fix $m=5$. The $x$-axis is set as time $t$ from $t=0$ to $t=10^4$, and the $y$-axis is set as the mean degree $<k>$. We observe the evolution process of $<k>$s with $\lambda$s$=[2,3,4,5]$. The black line is the theoretical expected degree of each evolving networks. In (b) and (e), we fix $\lambda=3$ and set $m$s$=[4,6,8,10]$. The $x$-axis and $y$-axis are defined the same as (a) and (d). Dashed lines present the theoretical expectation. In (c) and (f), we obtain the average mean degree from $t=10^3$ to $t=10^4$ for $m$s$=[4,6,8,10]$. The $x$-axis is set as the input rate $\lambda\in[2,5]$, where the simulations are performed with the interval $0.2$. The $y$-axis is defined the same as above. (Color online)
• Figure 5: Evolution of the random drift for BDEN-uc and BDEN-pa. (a) $\alpha=0.05$. (b) $\alpha=0.25$. In this figure, we set $\lambda=3$ and $m=4$ to show an evolution process of $A(t)$ for the proposed random drift mechanism. The orange and blue curves present the evolution processes that converge to the pure aware and unaware states separately. Note that for each evolution process, we stop collecting the number of aware individuals once $N(t)=A(t)$ or $N(t)=U(t)$, i.e., there is only one state among the population. The initial network is set as a complete graph with $N(0)=30$ vertices and $A(0)=1$ information invader. For each $\alpha$, we perform $1500$ independent and repeated experiments and obtain the above evolution results for both BDEN-uc and BDEN-pa. (Color online)

Theorems & Definitions (15)

• Definition 1
• Definition 2
• Theorem 1
• Proof
• Remark 1
• Definition 3
• Definition 4
• Theorem 2
• Proof
• Theorem 3